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3. The Neumann Laplacian

    1. Optimizers for Neumann eigenvalues among convex sets of a fixed perimeter

      Problem 3.1.

      [A. Henrot] Prove the existence of optimizers for $$ \sup\{\mu_k(\Omega)\colon \Omega\subset \mathbb{R}^n \mbox{ convex, }P(\Omega)=P_0\} $$ and $$ \inf\{\mu_k(\Omega)\colon \Omega\subset \mathbb{R}^n \mbox{ convex, }P(\Omega)=P_0\}. $$
          For $k=1$, $n=2$ and convex $\Omega$, R. Laugesen and B. Siudeja have conjectured that $P(\Omega)^2\mu_1(\Omega)\leq 16\pi^2$, where equality is attained if $\Omega$ is either a square or an equilateral triangle (open problem 6.66 in [MR3681143]).
        • Maximizing the ratio of Neumann eigenvalues among convex domains

          Conjecture 3.2.

          [A. Henrot] For any convex $\Omega\subset \mathbb{R}^n$, it holds that
          1. $\mu_2(\Omega)/\mu_1(\Omega)\leq 4$.
          2. $\mu_k(\Omega)/\mu_1(\Omega)\leq k^2$.
              The two-dimensional version of the above conjecture is formulated by M. Ashbaugh and R. Benguria in [MR1215424]. Partial analytic result again in two dimensions is obtained by P. Antunes and A. Henrot in [MR2795792].
            • Hot spots conjecture

                  This conjecture is often attributed to J. Rauch, who has formulated it in terms of large-time behaviuor of the heat semigroup generated by the Neumann Laplacian. On the qualitative level, this conjecture manifests as follows: for an insulated flat piece of metal with an arbitrary initial temperature distribution, given enough time, the hottest point on the metal will lie on its boundary.

              Conjecture 3.3.

              The first non-trivial Neumann eigenfunction does not have global extrema in the interior of $\Omega$ if
              1. $\Omega\subset \mathbb{R}^2$ is simply connected.
              2. $\Omega\subset \mathbb{R}^n$ ($n \ge 3$) is convex.
                  Positive results concern various special classes of domains such as triangles [arxiv:1802.01800], $\mathsf{Lip}$-domains [MR2051611], and thin curved strips [MR3912674]. A counterexample in the class of non-simply-connected planar domains is constructed in [MR1680567].
                • Directional hot spots conjecture

                      Let $\Omega \subset \mathbb{R}^2$ be a centrally symmetric bounded convex domain and let $u$ denote the eigenfunction corresponding to the first non-trivial Neumann eigenvalue.

                  Problem 3.4.

                  [T. Beck] Can one always find a direction $\mathbf{e}\in\mathbb{R}^2$ such that \begin{equation} \mathbf{e}\cdot \nabla u >0 \quad \mbox{in }\Omega? \end{equation}
                      An affirmative answer to the above question implies the hot spots conjecture for all centrally symmetric bounded convex planar domains. Under the assumption that $\Omega$ is symmetric with respect to both coordinate axes, the respective question has been affirmatively answered by D. Jerison and N. Nadirashvili in [MR1775736].

                      Cite this as: AimPL: Shape optimization with surface interactions, available at http://aimpl.org/shapesurface.